Optimal Resource Allocation over Networks via Lottery-Based Mechanisms
aa r X i v : . [ ec on . T H ] D ec Optimal Resource Allocation over Networks viaLottery-Based Mechanisms
Soham R. Phade and Venkat Anantharam *† Abstract
We show that, in a resource allocation problem, the ex ante aggre-gate utility of players with cumulative-prospect-theoretic preferencescan be increased over deterministic allocations by implementing lotter-ies. We formulate an optimization problem, called the system prob-lem, to find the optimal lottery allocation. The system problem exhibitsa two-layer structure comprised of a permutation profile and optimalallocations given the permutation profile. For any fixed permutationprofile, we provide a market-based mechanism to find the optimal allo-cations and prove the existence of equilibrium prices. We show that thesystem problem has a duality gap, in general, and that the primal prob-lem is NP-hard. We then consider a relaxation of the system problemand derive some qualitative features of the optimal lottery structure.
We consider the problem of congestion management in a network, and re-source allocation amongst heterogeneous users, in particular human agents,with varying preferences. This is a well-recognized problem in networkeconomics, with applications to transportation and telecommunication net-works, energy smart grids, information and financial networks, labor mar-kets and social networks, to name a few [20]. Market-based solutions haveproven to be very useful for this purpose, with varied mechanisms, such as * Research supported by the NSF Science and Technology Center grant CCF- 0939370:"Science of Information", the NSF grants ECCS-1343398, CNS-1527846 and CIF-1618145,and the William and Flora Hewlett Foundation supported Center for Long Term Cybersecurityat Berkeley. † The authors are with the Department of Electrical Engineering and Computer Science,University of California, Berkeley, Berkeley, CA 94720. [email protected],[email protected]
Do lotteries provide anadvantage over deterministic implementations? (ii)
If yes, then does there exista market-based mechanism to implement an optimum lottery?
In order to answer the first question we need to define our goal in allocat-ing resources. There is an extensive literature on the advantages of lotteries:Eckhoff [7] and Stone [25] hold that lotteries are used because of fairnessconcerns; Boyce [3] argues that lotteries are effective to reduce rent-seekingfrom speculators; Morgan [19] shows that lotteries are an effective way of fi-nancing public goods through voluntary funds, when the entity raising fundslacks tax power; Hylland and Zeckhauser [10] propose implementing lotter-ies to elicit honest preferences and allocate jobs efficiently. In all of theseworks, there is an underlying assumption, which is also one of the key rea-sons for the use of lotteries, that the goods to be allocated are indivisible.However, we notice lotteries being implemented even when the goods tobe allocated are divisible, for example in lottos and parimutuel betting. Inseveral experiments [21], it has been observed that lottery-based rewardsare more appealing than deterministic rewards of the same expected value,and thus provide an advantage in maximizing the desired influence on peo-ple’s behavior. We also observe several firms presenting lottery-based offersto incentivize customers into buying their products or using their services,and in return to improve their revenues. Thus, although lottery-based mech-anisms are being widely implemented, a theoretical understanding for thesame seems to be lacking. This is one of the motivations for this paper, whichaims to justify the use of lottery-based mechanisms, based on models comingfrom behavioral economics for how humans evaluate options.We take a utilitarian approach of maximizing the ex ante aggregate util-ity or the net happiness of the players. (See [1] and the references thereinfor the relation with other goals such as maximizing revenue.) We modeleach player’s utility using cumulative prospect theory (CPT), a frameworkpioneered by Tversky and Kahneman [27], which is believed, based on ex-tensive experimentation with human subjects [4], to form a better theorywith which to model human behavior when faced with prospects than isexpected utility theory (EUT) [28]. It is important to emphasize that CPTincludes EUT as a special case and therefore provides a strict generalizationof existing modeling techniques.CPT posits a probability weighting function that, along with the orderingof the allocation outcomes in a lottery, dictates the probabilistic sensitivity of aplayer (details in Section 2), a property that plays an important role in lotter-2es and gambling. As Boyce [3] points out, “it is the lure of getting the goodwithout having to pay for it that gives allocation by lottery its appeal.” Theprobability weighting function typically over-weights small probabilities andunder-weights large probabilities, and this captures the “lure” effect. CPTalso posits a reference point that divides the prospect outcomes into gainsand losses domains in order to model the loss aversion of the players. Inorder to focus on the effects of probabilistic sensitivity, and to avoid the com-plications resulting from reference point considerations, we assume that thereference point of all the players is equal to , and we consider prospects withonly nonnegative outcomes. This is, in fact, identical to the rank dependentutility (RDU) model [22].We will mainly be concerned with the framework considered in [13],that of throughput control in the internet with elastic traffic. However, thisframework is general enough to have applications to network resource allo-cation problems arising in several other domains. Kelly suggested that thethroughput allocation problem be posed as one of achieving maximum aggre-gate utility for the users. A market is proposed, in which each user submitsan amount she is willing to pay per unit time to the network based on ten-tative rates that she received from the network; the network accepts thesesubmitted amounts and determines the price of each network link. A useris then allocated a throughput in proportion to her submitted amount andinversely proportional to the sum of the prices of the links she wishes to use.Under certain assumptions, Kelly shows that there exist equilibrium pricesand throughput allocations, and that these allocations achieve maximum ag-gregate utility. Thus the overall system problem of maximizing aggregateutility is decomposed into a network problem and several user problems , onefor each individual user. Further, in [14], the authors have proposed twoclasses of algorithms which can be used to implement a relaxation of theabove optimization problem.Instead of allocating a single throughput, we consider allocating a prospect of throughputs to each user. Such a prospect consists of a finite set of through-puts and a probability assigned to each of these throughputs, with the in-terpretation that one of these throughputs would be realized with its cor-responding probability (see Section 2 for the definition). We then ask thequestion of finding the optimum allocation profile of prospects, one for eachuser, comprised of throughputs and associated probabilities for that user, thatmaximizes the aggregate utility of all the players, and is also feasible . An al-location profile of prospects for each user is said to be feasible if it can be im-plemented, i.e. there exists a probability distribution over feasible through-put allocations whose marginals for each player agree with their allocated3rospects.If all the players have EUT utility with concave utility function, as is typ-ically assumed to model risk-averseness, one can show that there exists afeasible deterministic allocation that achieves the optimum and hence thereis no need to consider lotteries. However, if the players’ utility is modeled byCPT, then one can improve over the best aggregate utility obtained throughdeterministic allocations.For example, Quiggin [23] considers the problem of distributing a fixedamount amongst several homogeneous players with RDU preferences. Heconcludes that, under certain conditions on players’ RDU preferences, theoptimum allocation system is a lottery scheme with a few large prizes and alarge number of small prizes, and is strictly preferred over distributing thetotal amount deterministically amongst the players. In Section 5, we extendthese results to network settings with heterogeneous players.In Section 2, we describe the network model, the lottery structure, andthe CPT model of player utility. We formulate an optimization problem,called the system problem, to find the optimum lottery scheme. The solu-tion of such an optimization problem, as explained in Section 2, exhibits alayered structure of finding a permutation profile , and corresponding feasi-ble throughput allocations. The permutation profile dictates in which orderthroughput allocations for each player are coupled together for network fea-sibility purposes. A player’s CPT value for her lottery allocation depends onlyon the order amongst her own throughputs, and not on the coupling with theother players.Given a permutation profile, the problem of finding optimum feasiblethroughput allocations is a convex programming problem, which we call thefixed-permutation system problem, and leads to a nice price mechanism. InSection 3, we prove the existence of equilibrium prices that decompose thefixed-permutation system problem into a network problem and several userproblems, one for each player, as in [13]. The prices can be interpreted asthe cost imposed on the players and can be implemented in several forms,such as waiting times in waiting-line auctions or first-come-first-served allo-cations [2, 26], delay or packet loss in the Internet TCP protocol [17, 15],efforts or resources invested by players in a contest [18, 6], or simply moneyor reward points.Finding the optimum permutation profile, on the other hand, is a non-convex problem. In Section 4, we study the duality gap in the system problemand consider a relaxation of the system problem by allowing the permuta-tions to be doubly stochastic matrices instead of restricting them to be per-mutation matrices. We show that strong duality holds in the relaxed system4roblem and so it has value equal to the dual of the original system problem(Theorem 4.2). We also consider the problem where link constraints holdin expectation, called the average system problem, and show that strong du-ality holds in this case and so it has value equal to the relaxed problem. InSection 5, we study the average system problem in further detail, and prove aresult on the structure of optimal lotteries. Example 4.3, establishes that theduality gap in the original system problem can be nonzero and Theorem 4.4shows that the primal system problem is NP-hard. In Section 6, we concludewith some open problems for future research. Consider a network with a set [ m ] = { , . . . , m } of resources or links and aset [ n ] = { , . . . , n } of users or players . Let c j > denote the finite capacity of link j ∈ [ m ] and let c := ( c j ) j ∈ [ m ] ∈ R m . (All vectors, unless otherwisespecified, will be treated as column vectors.) Each user i has a fixed route J i , which is a non-empty subset of [ m ] . Let A be an n × m matrix, where A ij = 1 if link j ∈ J i , and A ij = 0 otherwise. Let x := ( x i ) i ∈ [ n ] ∈ R n + denotean allocation profile where user i is allocated the throughput x i ≥ thatflows through the links in the route J i . We say that an allocation profile x isfeasible if it satisfies the capacity constraints of the network, i.e., A T x ≤ c ,where the inequality is coordinatewise. Let F denote the set of all feasibleallocation profiles. We assume that the network constraints are such that F is bounded, and hence a polytope.Instead of allocating a fixed throughput x i to player i ∈ [ n ] , we considerallocating her a lottery (or a prospect ) L i := { ( p i (1) , y i (1)) , . . . , ( p i ( k i ) , y i ( k i )) } , (2.1)where y i ( l i ) ≥ , l i ∈ [ k i ] , denotes a throughput and p i ( l i ) , l i ∈ [ k i ] , is theprobability with which throughput y i ( l i ) is allocated. We assume the lotteryto be exhaustive, i.e. P l i ∈ [ k i ] p i ( l i ) = 1 . (Note that we are allowed to have p i ( l i ) = 0 for some values of l i ∈ [ k i ] and y i ( l i ) = y i ( l i ) for some l i , l i ∈ [ k i ] .)Let L = ( L i , i ∈ [ n ]) denote a lottery profile , where each player i is allocatedlottery L i .We now describe the CPT model we use to measure the “utility” or “hap-piness” derived by each player from her lottery (for more details see [29]).Each player i is associated with a value function v i : R + → R + that is con-tinuous, differentiable, concave, and strictly increasing, and a probability5eighting function w i : [0 , → [0 , that is continuous, strictly increasingand satisfies w i (0) = 0 and w i (1) = 1 .For the prospect L i in (2.1), let π i : [ k i ] → [ k i ] be a permutation such that z i (1) ≥ z i (2) ≥ · · · ≥ z i ( k i ) , (2.2)and y i ( l i ) = z i ( π i ( l i )) for all l i ∈ [ k i ] . (2.3)The prospect L i can equivalently be written as L i = { (˜ p i (1) , z i (1)); . . . ; (˜ p i ( k i ) , z i ( k i )) } , where ˜ p i ( l i ) := p i ( π − i ( l i )) for all l i ∈ [ k i ] . The CPT value of prospect L i for player i is evaluated using the value function v i ( · ) and the probabilityweighting function w i ( · ) as follows: V i ( L i ) := k i X l i =1 d l i ( p i , π i ) v i ( z i ( l i )) , (2.4)where d l i ( p i , π i ) are the decision weights given by d ( p i , π i ) := w i (˜ p i (1)) and d l i ( p i , π i ) := w i (˜ p i (1) + · · · + ˜ p i ( l i )) − w i (˜ p i (1) + · · · + ˜ p i ( l i − , for < l i ≤ k i . Although the expression on the right in equation (2.4)depends on the permutation π i , one can check that the formula evaluates tothe same value V i ( L i ) as long as π i satisfies (2.2) and (2.3). The CPT valueof prospect L i , can equivalently be written as V i ( L i ) = k i X l i =1 w i (cid:0) l i X s i =1 ˜ p i ( s i ) (cid:1) [ v i ( z i ( l i )) − v i ( z i ( l i + 1)))] , where z i ( k i +1) := 0 . Thus the lowest allocation z i ( k i ) is weighted by w i (1) =1 , and every increment in the value of the allocations, v i ( z i ( l i )) − v i ( z i ( l i +1)) , ∀ l i ∈ [ k i − , is weighted by the probability weighting function of theprobability of receiving an allocation at least equal to z i ( l i ) .For any finite set S , let ∆( S ) denote the standard simplex of all probabil-ity distributions on the set S , i.e., ∆( S ) := { ( p ( s ) , s ∈ S ) | p ( s ) ≥ ∀ s ∈ S, X s ∈ S p ( s ) = 1 } . Thus p i := ( p i ( l i )) l i ∈ [ k i ] ∈ ∆([ k i ]) . We say that a lottery profile L is feasi-ble if there exists a joint distribution p ∈ ∆( Q i [ k i ]) such that the followingconditions are satisfied: 6i) The marginal distributions agree with L i for all players i , i.e. P l − i p ( l i , l − i ) = p i ( l i ) for all l i ∈ [ k i ] , where l − i in the summation ranges over values in Q i ′ = i [ k i ′ ] .(ii) For each ( l i ) i ∈ [ n ] ∈ Q i [ k i ] in the support of the distribution p (i.e. p (( l i ) i ∈ [ n ] ) > ), the allocation profile ( y i ( l i )) i ∈ [ n ] is feasible.The distribution p and the throughputs ( y i ( l i ) , i ∈ [ n ] , l i ∈ [ k i ]) of a feasi-ble lottery profile together define a lottery scheme . In the following, we re-strict our attention to specific types of lottery schemes, wherein the networkimplements with equal probability one of the k allocation profiles y ( l ) :=( y i ( l )) i ∈ [ n ] ∈ R n + , for l ∈ [ k ] . Let [ k ] = { , . . . , k } denote the set of outcomes ,where allocation profile y i ( l ) is implemented if outcome l occurs. Clearly,such a scheme is feasible iff each of the allocation profiles y ( l ) , ∀ l ∈ [ k ] be-longs to F . Player i thus faces the prospect L i = { (1 /k, y i ( l )) } kl =1 and such alottery scheme is completely characterized by the tuple y := ( y i ( l ) , i ∈ [ n ] , l ∈ [ k ]) . By taking k large enough, any lottery scheme can be approximated bysuch a scheme.Let y i := ( y i ( l )) l ∈ [ k ] ∈ R k + . Let z i := ( z i ( l )) l ∈ [ k ] ∈ R k + be a vector and π i : [ k ] → [ k ] be a permutation such that z i (1) ≥ z i (2) ≥ · · · ≥ z i ( k ) , and y i ( l ) = z i ( π i ( l )) for all l ∈ [ k ] . Note that y i is completely characterized by π i and z i . Then player i ’s CPTvalue will be V i ( L i ) = k X l =1 h i ( l ) v i ( z i ( l )) , where h i ( l ) := w i ( l/k ) − w i (( l − /k ) for l ∈ [ k ] . Let h i := ( h i ( l )) l ∈ [ k ] ∈ R k + .Note that h i ( l ) > for all i, l , since the weighting functions are assumed tobe strictly increasing.Looking at the lottery scheme y in terms of individual allocation profiles z i and permutations π i for all players i ∈ [ n ] , allows us to separate thosefeatures of y that affect individual preferences and those that pertain to thenetwork implementation. We will later see that the problem of optimizingaggregate utility can be decomposed into two layers: (i) a convex problemthat optimizes over resource allocations, and (ii) a non-convex problem thatfinds the optimal permutation profile.7et z := ( z i ( l ) , i ∈ [ n ] , l ∈ [ k ]) , π := ( π i , i ∈ [ n ]) , h := ( h i ( l ) , i ∈ [ n ] , l ∈ [ k ]) and v := ( v i ( · ) , i ∈ [ n ]) . Let S k denote the set of all permutations of [ k ] .The problem of optimizing aggregate utility P i V i ( L i ) subject to the lotteryscheme being feasible, can be formulated as follows:SYS [ z, π ; h, v, A, c ] Maximize n X i =1 k X l =1 h i ( l ) v i ( z i ( l )) subject to X i ∈ R j z i ( π i ( l )) ≤ c j , ∀ j ∈ [ m ] , ∀ l ∈ [ k ] ,z i ( l ) ≥ z i ( l + 1) , ∀ i ∈ [ n ] , ∀ l ∈ [ k ] ,π i ∈ S k , ∀ i ∈ [ n ] . Here R j := { i ∈ [ n ] | j ∈ J i } is the set of all players whose route uses link j .We set z i ( k + 1) = 0 for all i , and the z i ( k + 1) are not treated as variables.This takes care of the condition z i ( l ) ≥ for all i ∈ [ n ] , l ∈ [ k ] . The system problem SYS [ z, π ; h, v, A, c ] optimizes over z and π . In this sec-tion we fix π i ∈ S k for all i and optimize over z . Let us denote this fixed-permutation system problem by SYS_FIX [ z ; π, h, v, A, c ] .SYS_FIX [ z ; π, h, v, A, c ] Maximize n X i =1 k X l =1 h i ( l ) v i ( z i ( l )) subject to X i ∈ R j z i ( π i ( l )) ≤ c j , ∀ j ∈ [ m ] , ∀ l ∈ [ k ] ,z i ( l ) ≥ z i ( l + 1) , ∀ i ∈ [ n ] , ∀ l ∈ [ k ] . (In contrast with SYS ( z, π ; . . . ) , in SYS_FIX ( z ; π, . . . ) , the permutation π isthought of as being fixed.) Since v i ( · ) is assumed to be a concave functionand h i ( l ) > for all i, l , this problem has a concave objective function withlinear constraints. For all j ∈ [ m ] , l ∈ [ k ] , let λ j ( l ) ≥ be the dual variablescorresponding to the constraints P i ∈ R j z i ( π i ( l )) ≤ c j respectively, and forall i ∈ [ n ] , l ∈ [ k ] , let α i ( l ) ≥ be the dual variables corresponding to theconstraints z i ( l ) ≥ z i ( l + 1) respectively. Let λ := ( λ j ( l ) , j ∈ [ m ] , l ∈ [ k ]) and8 := ( α i ( t ) , i ∈ [ n ] , l ∈ [ k ]) . Then the Lagrangian for the fixed-permutationsystem problem SYS_FIX [ z ; π, h, v, A, c ] can be written as follows: L ( z ; α, λ ) := n X i =1 k X l =1 h i ( l ) v i ( z i ( l ))+ n X i =1 k X l =1 α i ( l )[ z i ( l ) − z i ( l + 1)] + m X j =1 k X l =1 λ j ( l )[ c j − X i ∈ R j z i ( π i ( l ))]= n X i =1 k X l =1 h i ( l ) v i ( z i ( l )) + ( α i ( l ) − α i ( l − z i ( l ) − X j ∈ J i λ j ( π − i ( l )) z i ( l ) + m X j =1 k X l =1 λ j ( l ) c j , where α i (0) = 0 for all i ∈ [ n ] . Differentiating the Lagrangian with respectto z i ( l ) we get, ∂ L ( z ; α, λ ) ∂z i ( l ) = h i ( l ) v ′ i ( z i ( l )) + α i ( l ) − α i ( l − − X j ∈ J i λ j ( π − i ( l )) . Let ρ i ( l ) := X j ∈ J i λ j ( π − i ( l )) , (3.1)for all i ∈ [ n ] , l ∈ [ k ] . This can be interpreted as the price per unit throughputfor player i for her l -th largest allocation z i ( l ) . The price of the lottery z i forplayer i is given by P kl =1 ρ i ( l ) z i ( l ) , or equivalently, k X l =1 r i ( l ) [ z i ( l ) − z i ( l + 1)] , where r i ( l ) := l X s =1 ρ i ( s ) , for all l ∈ [ k ] . (3.2)For l ∈ [ k − , α i ( l ) can be interpreted as a transfer of a nonnegative pricefor player i from her l -th largest allocation to her ( l + 1 )-th largest allocation.Since the allocation z i ( l + 1) cannot be greater than the allocation z i ( l ) , thereis a subsidy of α i ( l ) in the price of z i ( l ) and an equal surcharge of α i ( l ) in the9rice of z i ( l + 1) . This subsidy and surcharge is nonzero (and hence positive)only if the constraint is binding, i.e. z i ( l ) = z i ( l + 1) . On the other hand, α i ( k ) is a subsidy in price given to player i for her lowest allocation, sinceshe cannot be charged anything higher than the marginal utility at her zeroallocation.Let h i := ( h i ( l )) l ∈ [ k ] ∈ R k + . Consider the following user problem forplayer i :USER [ m i ; r i , h i , v i ] Maximize k X l =1 h i ( l ) v i k X s = l m i ( s ) r i ( s ) ! − k X l =1 m i ( l ) subject to m i ( l ) ≥ , ∀ l ∈ [ k ] , (3.3)where r i := ( r i ( l ) , l ∈ [ k ]) is a vector of rates such that < r i (1) ≤ r i (2) ≤ · · · ≤ r i ( k ) . (3.4)We can interpret this as follows: User i is charged rate r i ( k ) for her lowestallocation δ i ( k ) := z i ( k ) . Let m i ( k ) denote the budget spent on the lowestallocation and hence m i ( k ) = r i ( k ) δ i ( k ) . For ≤ l < k , she is charged rate r i ( l ) for the additional allocation δ i ( l ) := z i ( l ) − z i ( l + 1) , beyond z i ( l + 1) up to the next lowest allocation z i ( l ) . Let m i ( l ) denote the budget spent on l -th additional allocation and hence m i ( l ) = r i ( l ) δ i ( l ) .Let m := ( m i ( l ) , i ∈ [ n ] , l ∈ [ k ]) and δ := ( δ i ( l ) , i ∈ [ n ] , l ∈ [ k ]) . Considerthe following network problem:NET [ δ ; m, π, A, c ] Maximize n X i =1 k X l =1 m i ( l ) log( δ i ( l )) subject to δ i ( l ) ≥ , ∀ i, ∀ l, X i ∈ R j k X s = π i ( l ) δ i ( s ) ≤ c j , ∀ j, ∀ l. This is the well known Eisenberg-Gale convex program [8] and it can besolved efficiently. Kelly et al. [14] proposed continuous time algorithms forfinding equilibrium prices and allocations. For results on polynomial timealgorithms for these problems see [11, 5]. We have the following decompo-sition result: 10 heorem 3.1.
For any fixed π , there exist equilibrium parameters r ∗ , m ∗ , δ ∗ and z ∗ such that(i) for each player i , m ∗ i solves the user problem USER [ m i ; r ∗ i , h i , v i ] ,(ii) δ ∗ solves the network problem NET [ δ ; m ∗ , π, A, c ] ,(iii) m ∗ i ( l ) = δ ∗ i ( l ) r ∗ i ( l ) for all i, l ,(iv) δ ∗ i ( l ) = z ∗ i ( l ) − z ∗ i ( l + 1) for all i, l , and(v) z ∗ solves the fixed-permutation system problem SYS_FIX [ z ; π, h, v, A, c ] .Proof. Since SYS_FIX [ z ; π, h, v, A, c ] is a convex optimization problem, weknow that there exist z ∗ = ( z ∗ i ( l ) , i ∈ [ n ] , l ∈ [ k ]) , α ∗ = ( α ∗ i ( l ) , i ∈ [ n ] , l ∈ [ k ]) and λ ∗ = ( λ ∗ j ( l ) , j ∈ [ m ] , l ∈ [ k ]) such that h i ( l ) v ′ i ( z ∗ i ( l )) = ρ ∗ i ( l ) − α ∗ i ( l ) + α ∗ i ( l − , ∀ i, ∀ l, (3.5) z ∗ i ( l ) ≥ z ∗ i ( l + 1) , α ∗ i ( l ) ≥ , α ∗ i ( l )( z ∗ i ( l ) − z ∗ i ( l + 1)) = 0 , ∀ i, ∀ l, (3.6) X i ∈ R j z ∗ i ( π i ( l )) ≤ c j , λ ∗ j ( l ) ≥ , λ ∗ j ( l )[ c j − X i ∈ R j z ∗ i ( π i ( l ))] = 0 , ∀ j, ∀ l, (3.7)where ρ ∗ i ( l ) := X j ∈ J i λ ∗ j ( π − i ( l )) , and such that z ∗ solves the fixed-permutation system problem SYS_FIX [ z ; π, h, v, A, c ] .Hence statement (v) holds for this choice of z ∗ . Let r ∗ i ( l ) := P ls =1 ρ ∗ i ( s ) , δ ∗ i ( l ) := z ∗ i ( l ) − z ∗ i ( l + 1) and m ∗ i ( l ) := δ ∗ i ( l ) r ∗ i ( l ) for all i, l . From (3.5), wehave ρ ∗ i (1) > , because v i ( · ) is strictly increasing, h i (1) > and α ∗ i (0) = 0 .Thus, the rate vector r ∗ i satisfies (3.4) for all i . Also note that, by construction,the vectors r ∗ , δ ∗ , z ∗ and m ∗ satisfy statements (iii) and (iv) of the theorem.We now show that statement (i) holds. Fix a player i . Observe thatthe user problem USER [ m i ; r ∗ i , h i , v i ] has a concave objective function since h i ( l ) > , r ∗ i ( l ) > and v i ( · ) is concave. Differentiating the objective functionof the user problem USER [ m i ; r ∗ i , h i , v i ] with respect to m i ( l ) at m i = m ∗ i , weget l X s =1 h i ( s ) v ′ i ( z ∗ i ( s )) r ∗ i ( l ) − . l X s =1 h i ( s ) v ′ i ( z ∗ i ( l )) = l X s =1 ρ ∗ i ( s ) − α ∗ i ( l )= r ∗ i ( l ) − α ∗ i ( l ) . Thus, l X s =1 h i ( s ) v ′ i ( z ∗ i ( s )) r ∗ i ( l ) − − α ∗ i ( l ) r ∗ i ( l ) ≤ , where equality holds iff α ∗ i ( l ) = 0 . If m ∗ i ( l ) > , then δ ∗ i ( l ) > and hence,by (3.6), α ∗ i ( l ) = 0 . Since this holds for all l ∈ [ k ] , these are precisely theconditions necessary and sufficient for the optimality of m ∗ i in the problemof user i , it being a convex problem.We now show that statement (ii) holds. The Lagrangian correspondingto the network problem NET [ δ ; m ∗ , π, A, c ] can be written as follows: L ( δ ; µ ) = X i k X l =1 m ∗ i ( l ) log( δ i ( l )) + X j X l µ j ( l ) c j − X i ∈ R j k X s = π i ( l ) δ i ( s ) , where µ j ( l ) is the dual variable corresponding to the link constraint and µ :=( µ j ( l ) , j ∈ [ n ] , l ∈ [ k ]) . Let µ j ( l ) = λ ∗ j ( l ) . If m ∗ i ( l ) > , then δ ∗ i ( l ) > , anddifferentiating the Lagrangian with respect to δ i ( l ) at δ ∗ i ( l ) , we get ∂ L ( δ ; λ ∗ ) ∂δ i ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ i ( l )= δ ∗ i ( l ) = m ∗ i ( l ) δ ∗ i ( l ) − X j ∈ J i l X s =1 λ ∗ j ( π − i ( s ))= r ∗ i ( l ) − l X s =1 ρ ∗ i ( s ) = 0 . If m ∗ i ( l ) = 0 , then ∂ L ( δ ; λ ∗ ) ∂δ i ( l ) (cid:12)(cid:12) δ i ( l )= δ ∗ i ( l ) ≤ since λ ∗ j ( l ) ≥ for all j, l . Further,from (3.7), we have λ ∗ j ( l ) h c j − P i ∈ R j P ks = π i ( l ) δ ∗ i ( s ) i = 0 for all j, l . Thus, δ ∗ solves the network problem NET [ δ ; m ∗ , π, A, c ] .Thus the fixed-permutation system problem can be decomposed into userproblems – one for each player – and a network problem, for any fixed per-mutation profile π . Similar to the framework in [14], we have an iterativeprocess as follows: The network presents each user i with a rate vector r i .12ach user solves the user problem USER [ m i ; r i , h i , v i ] , and submits their bud-get vector m i , The network collects these budget vectors ( m i ) i ∈ [ n ] and solvesthe network problem NET [ δ ; m ∗ , π, A, c ] to get the corresponding allocation z (which can be computed from the incremental allocations δ ) and the dualvariables λ . The network then computes the rate vectors corresponding toeach user from these dual variables as given by (3.1) and (3.2) and presents itto the users as updated rates. Theorem 3.1 shows that the fixed-permutationsystem problem of maximizing the aggregate utility is solved at the equilib-rium of the above iterative process. If the value functions v i ( · ) are strictlyconcave, then one can show that the optimal lottery allocation z ∗ for thefixed-permutation system problem is unique. However, the dual variables λ ,and hence the rates r i , ∀ i , need not be unique. Nonetheless, if one uses thecontinuous-time algorithm proposed in [14] to solve the network problem,then a similar analysis as in [14], based on Lyapunov stability, shows thatthe above iterative process converges to the equilibrium lottery allocation z ∗ .One of the permutation profiles, say π ∗ , solves the system problem. Inthe next section, we explore this in more detail. However, it is interesting tonote that, for any fixed permutation profile π , any deterministic solution isa special case of the lottery scheme y with permutation profile π . Thus, itis guaranteed that the solution of the fixed-permutation system problem forany permutation profile π is at least as good as any deterministic allocation.Here is a simple example, where a lottery-based allocation leads to strictimprovement over deterministic allocations. Example . Consider a network with n players and a single link with capac-ity c . Let n = 10 and c = 10 . For all players i , we employ the value functionsand weighting functions suggested by Kahneman and Tversky [27], given by v i ( x i ) = x β i i , β i ∈ [0 , , and w i ( p i ) = p γ i i ( p γ i + (1 − p ) γ i ) /γ i , γ i ∈ (0 , , respectively. We take β i = 0 . and γ i = 0 . for all i ∈ [ n ] . These parameterswere reported as the best fits to the empirical data in [27]. The probabilityweighting function is displayed in Figure 1.By symmetry and concavity of the value function v i ( · ) , the optimal deter-ministic allocation is given by allocating c/n to each player i . The aggregateutility for this allocation is n ∗ v ( c/n ) = 10 .Now consider the following lottery allocation: Let k = n = 10 . Let π i ( l ) − l + i ( mod k ) for all i ∈ [ n ] and l ∈ [ k ] . Let x ∈ [ c/n, c ] and z i (1) = x . . . . . . . . p i w i ( p i ) Figure 1: Probability weighting functionfor all i ∈ [ n ] and z i ( l ) = ( c − x ) / ( n − for all i ∈ [ n ] and l = 2 , . . . , k .Note that this is a feasible lottery allocation. Such a lottery scheme can beinterpreted as follows: Select a “winning” player uniformly at random fromall the players. Allocate her a reward x and equally distribute the remainingreward c − x amongst the rest of the players. The ex ante aggregate utilityis given by n ∗ [ w (1 /n ) v ( x ) + (1 − w (1 /n )) v (( c − x ) / ( n − . This function achieves its maximum equal to . at x = 9 . . Thus, theabove proposed lottery improves the aggregate utility over any deterministicallocation. The optimum lottery allocation is at least as good as . . The system problem SYS [ z, π ; h, v, A, c ] can equivalently be formulated as max π i ∈ S k ∀ i,z : z i ( l ) ≥ z i ( l +1) ∀ i,l min λ ≥ n X i =1 k X l =1 h i ( l ) v i ( z i ( l ))+ m X j =1 k X l =1 λ j ( l ) c j − X i ∈ R j z i ( π i ( l )) . (I)14et W ps denote the value of this problem. It is equal to the optimum value ofthe system problem SYS [ z, π ; h, v, A, c ] . By interchanging the max and min ,we obtain the following dual problem: min λ ≥ max π i ∈ S k ∀ i,z : z i ( l ) ≥ z i ( l +1) ∀ i,l n X i =1 k X l =1 h i ( l ) v i ( z i ( l ))+ m X j =1 k X l =1 λ j ( l ) c j − X i ∈ R j z i ( π i ( l )) . (II)Let W ds denote the value of this dual problem. By weak duality, we know that W ps ≤ W ds . For a fixed λ ≥ and a fixed z that satisfies z i ( l ) ≥ z i ( l + 1) , ∀ i, l ,the optimum permutation profile π in the dual problem (II) should minimize m X j =1 k X l =1 λ j ( l ) X i ∈ R j z i ( π i ( l )) , which equals X i X l ˆ ρ i ( l ) z i ( π i ( l )) , Here ˆ ρ i ( l ) := P j ∈ J i λ j ( l ) , is the price per unit allocation for player i underoutcome l . Since the numbers z i ( l ) are ordered in descending order, anyoptimal permutation π i must satisfy ˆ ρ i ( π − i (1)) ≤ ˆ ρ i ( π − i (2)) ≤ · · · ≤ ˆ ρ i ( π − i ( k )) . (4.1)In other words, any optimal permutation profile π of the dual problem (II)must allocate throughputs in the order opposite to that of the prices ˆ ρ i ( l ) . Lemma 4.1.
If strong duality holds between the problems (I) and (II) , thenany optimum permutation profile π ∗ satisfies (4.1) for all i . We prove this lemma in Appendix A. In general, there is a non-zero du-ality gap between the problems (I) and (II) (see Example 4.3 for such anexample where the optimum permutation profile π ∗ does not satisfy (4.1)).The permutation π i can be represented by a k × k permutation matrix M i , where M i ( s, t ) = 1 if π i ( s ) = t and M i ( s, t ) = 0 otherwise, for s, t ∈ [ k ] .The network constraints P i ∈ R j z i ( π i ( l )) ≤ c j , ∀ l ∈ [ k ] , can equivalently bewritten as P i ∈ R j M i z i ≤ c j , where denotes a vector of appropriate sizewith all its elements equal to , and the inequality is coordinatewise. A15ossible relaxation of the system problem is to consider doubly stochasticmatrices M i instead of restricting them to be permutation matrices. A matrixis said to be doubly stochastic if all its entries are nonnegative and each rowand column sums up to . A permutation matrix is hence a doubly stochasticmatrix. Let Ω k denote the set of all doubly stochastic k × k matrices and let Ω ∗ k denote the set of all k × k permutation matrices.Let M = ( M i , i ∈ [ n ]) denote a profile of doubly stochastic matrices. Therelaxed system problem can then be written as follows:SYS_REL [ z, M ; h, v, A, c ] Maximize n X i =1 k X l =1 h i ( l ) v i ( z i ( l )) subject to X i ∈ R j M i z i ≤ c j , ∀ j,z i ( l ) ≥ z i ( l + 1) , ∀ i, ∀ l,M i ∈ Ω k , ∀ i. Then the corresponding primal problem can be written as follows: max M i ∈ Ω k ∀ i,z : z i ( l ) ≥ z i ( l +1) ∀ l, ∀ i min λ j ≥ , ∀ j X i X l h i ( l ) v i ( z i ( l ))+ X j λ Tj c j − X i ∈ R j M i z i . (III)where λ j = ( λ j ( l )) l ∈ [ k ] ∈ R k + . Let W pr denote the value of this problem.Interchanging min and max we get the corresponding dual: min λ j ≥ , ∀ j max M i ∈ Ω k ∀ i,z : z i ( l ) ≥ z i ( l +1) ∀ l, ∀ i X i X l h i ( l ) v i ( z i ( l ))+ X j λ Tj c j − X i ∈ R j M i z i . (IV)Let W dr denote the value of this problem. If the link constraints in the relaxedsystem problem hold then k X i ∈ R j k X l =1 z i ( l ) = 1 k X i ∈ R j T M i z i ≤ k T c j = c j . (4.2)16his inequality essentially says that the link constraints should hold in ex-pectation. Thus we have the following average system problem:SYS_AVG [ z ; h, v, A, c ] Maximize n X i =1 k X l =1 h i ( l ) v i ( z i ( l )) subject to X i ∈ R j k k X l =1 z i ( l ) ≤ c j , ∀ j,z i ( l ) ≥ z i ( l + 1) , ∀ i, ∀ l, with its corresponding primal problem: max z : z i ( l ) ≥ z i ( l +1) ∀ l, ∀ i min ¯ λ j ≥ , ∀ j X i X l h i ( l ) v i ( z i ( l ))+ X j ¯ λ j c j − X i ∈ R j k k X l =1 z i ( l ) , (V)and the dual problem: min ¯ λ j ≥ , ∀ j max z : z i ( l ) ≥ z i ( l +1) ∀ l, ∀ i X i X l h i ( l ) v i ( z i ( l ))+ X j ¯ λ j c j − X i ∈ R j k k X l =1 z i ( l ) , (VI)where ¯ λ j ∈ R are the dual variables corresponding to the link constraints.Let W pa and W da denote the values of these primal and dual problems re-spectively.Then we have the following relation: Theorem 4.2.
For any system problem defined by h, v, A and c , we have W ps ≤ W pr = W pa = W da = W dr = W ds . Proof.
As observed earlier, if we replace the condition M i ∈ Ω k in the primalrelaxed problem (III) with the condition M i ∈ Ω ∗ k , we get the primal systemproblem (I). Since Ω ∗ k ⊂ Ω k , we have W ps ≤ W pr .The constraint P i ∈ R j k P kl =1 z i ( l ) ≤ c j can equivalently be written as X i ∈ R j ¯ M i z i ≤ c j , j , where ¯ M i is the matrix with all its entries equal to /k . Thus, ifwe replace M i ∈ Ω k in the primal relaxed problem (III) with ¯ M i , we getthe primal average problem (V). This implies that W pa ≤ W pr . However,as observed earlier in Equation (4.2), if for some fixed allocations z the linkconstraints are satisfied with respect to any doubly stochastic matrix M i , thenthey are also satisfied with respect to ¯ M i . Thus the maximum of the averagesystem problem SYS_AVG [ z ; h, v, A, c ] is at least as much as the maximumof the relaxed system problem SYS_REL [ h, v, A, c ] . Since the maximum ofthe relaxed system problem is equal to the value of its corresponding primalproblem, we get W pa ≥ W pr . Thus, we have established that W pr = W pa .The average system problem has a concave objective function with linearconstraints. Thus, strong duality holds, and we get W pa = W da .We now show that W da = W dr . From W pr ≤ W dr and W pr = W pa = W da ,we get W da ≤ W dr . Suppose λ j = ¯ λ j . Then the objective function of therelaxed dual problem (IV), n X i =1 k X l =1 h i ( l ) v i ( z i ( l )) + m X j =1 λ Tj c j − X i ∈ R j M i z i = n X i =1 k X l =1 h i ( l ) v i ( z i ( l )) + m X j =1 ¯ λ j c j − X i ∈ R j k k X l =1 z i ( l ) , equals the objective function of the dual average problem (V). This impliesthat W dr ≤ W da . This established that W da = W dr .Since any doubly stochastic matrix M i is a convex combination of per-mutation matrices by the Birkhoff-von Neumann theorem, in the dual prob-lem (IV), for any fixed λ j , z i , the optimum can be achieved by a permutationmatrix. This established that W dr = W ds .This completes the proof.Thus the duality gap is a manifestation of the “hard” link constraints. Inthe proof of the above theorem we saw that the relaxed problem is “equiva-lent” to the average problem and strong duality holds for this relaxation. Wewill later study the average problem in further detail (Section 5).We observed earlier in Lemma 4.1 that if strong duality holds in the sys-tem problem, then the optimum permutation profile π ∗ satisfies (4.1). Con-sider a simple example of two players sharing a single link. Suppose that,at the optimum, λ ( l ) are the prices for l ∈ [ k ] corresponding to this linkunder the different outcomes, and suppose not all of these are equal. Then18he optimum permutation profile of the dual problem will align both play-ers’ allocations in the same order, i.e. the high allocations of player will bealigned with the high allocations of player . However, we can directly seefrom the system problem that an optimum π ∗ should align the two players’allocations in opposite order. The following example builds on this observa-tion and shows that strong duality need not hold for the system problem. Example . Consider the following example with two players { , } and asingle link with capacity . . Let k = 2 . Let the corresponding CPT charac-teristics of the two players be as follows: h (1) = 13 , h (2) = 23 ,h (1) = 56 , h (2) = 16 ,v ( x ) = log( x + 0 .
05) + 3 , v ( x ) = 2 log( x + 0 .
05) + 3( x + 0 . . For this problem, it is easy to see that π = (1 , and π = (2 , is an optimalpermutation. Solving the fixed-permutation system problem with respect tothis permutation we get optimal value equal to . . The correspondingvariable values are z (1) = y (1) = 1 . , z (2) = y (2) = 0 . ,z (1) = y (2) = 1 . , z (2) = y (1) = 0 . , and the dual variable values are λ (1) = 16 , λ (2) = 23 , and α i ( l ) = 0 , for i = 1 , , l = 1 , . One can check that these satisfy the KKTconditions.Let us now evaluate the value of the dual problem (II). By symmetry, wecan assume without loss of generality that λ (1) ≤ λ (2) . As a result, optimalpermutations for the dual problem are given by π = π = (1 , . For fixed19 (1) and λ (2) , we solve the following optimization problem: max z (1) ≥ z (2) ≥ z (1) ≥ z (2) ≥
13 log( z (1) + 0 .
05) + 23 log( z (2) + 0 . (cid:20) z (1) + 0 .
05) + 3( z (1) + 0 . (cid:21) + 16 (cid:20) z (2) + 0 .
05) + 3( z (2) + 0 . (cid:21) − λ (1)[ z (1) + z (1)] − λ (2)[ z (2) + z (2)]+ 2 . λ (1) + λ (2)] + 6 . (VII)If λ (1) ≤ . , then the value of the problem (VII) is equal to ∞ (let z (1) → ∞ ). If λ (1) > . (and hence λ (2) > . because λ (2) ≥ λ (2) ),then we observe that the effective domain of maximization in the problem(VII) is compact and problem (VII) has a finite value. Hence it is enough toconsider λ (1) > . . At the optimum there exist α (1) , α (2) , α (1) , α (2) ≥ such that λ (1) = 13 1 z (1) + 0 .
05 + α (1) ,λ (2) = 23 1 z (2) + 0 . − α (1) + α (2) ,λ (1) = 13 1 z (1) + 0 .
05 + 12 + α (1) ,λ (2) = 115 1 z (2) + 0 .
05 + 110 − α (1) + α (2) , and α (1)[ z (1) − z (2)] = 0 , α (2) z (2) = 0 ,α (1)[ z (1) − z (2)] = 0 , α (2) z (2) = 0 . We now consider each of the sixteen ( × ) cases based on whether theinequalities z i ( l ) ≥ z i ( l + 1) for i = 1 , and l = 1 , , hold strictly or not. Case A1 ( z (1) = 0 , z (2) = 0 ) . Then λ (1) ≥ / . . Case B1 ( z (1) > , z (2) = 0 ) . Then λ (1) < / . , λ (2) ≥ / . ,and α (1) = 0 , z (1) = 13 λ (1) − . . ase C1 ( z (1) = z (2) > ) . Then λ (2) / ≤ λ (1) ≤ λ (2) , λ (1) + λ (2) < / . , and α (1) = 2 λ (1) − λ (2)3 , α (2) = 0 , z (1) = z (2) = 1 λ (1) + λ (2) − . . Case D1 ( z (1) > z (2) > ) . Then λ (1) < λ (2) / , < λ (1) < / . , < λ (2) < / . , and α (1) = 0 , α (2) = 0 , z (1) = 13 λ (1) − . , z (2) = 23 λ (2) − . . Case A2 ( z (1) = 0 , z (2) = 0 ) . Then λ (1) ≥ (1 / .
15) + 0 . . Case B2 ( z (1) > , z (2) = 0 ) . Then . < λ (1) < . / . , λ (2) ≥ (1 / .
75) + 0 . , and α (1) = 0 , z (1) = 26 λ (1) − − . . Case C2 ( z (1) = z (2) > ) . This is not possible since λ (1) ≤ λ (2) . Case D2 ( z (1) > z (2) > ) . Then . < λ (1) < . / . , . <λ (2) < . / . , and α (1) = 0 , α (2) = 0 , z (1) = 26 λ (1) − − . , z (2) = 230 λ (2) − − . . If case A1 or case A2 holds, then λ (1) ≥ / . . For any fixed λ (1) , λ (2) ,by choosing z i ( l ) , i = 1 , , l = 1 , small enough (respecting the conditionsimposed by the corresponding cases), we get that the value of problem (VII)is greater than or equal to . ∗ (1 / .
15) = 19 . > . . Similarly, if caseB1 holds, then λ (2) ≥ / . , and we get that the value of problem (VII) isgreater than or equal to . ∗ (2 / .
15) = 38 . > . . For the remaining cases, substituting the corresponding expressions for z i ( l ) , i = 1 , , l = 1 , in the objective function (VII) and evaluating the optimum over feasible pairs ( λ (1) , λ (2)) for each pair of cases { C1,D1 } × {
B2,D2 } , the minimum isachieved for the case (C1,D2) and has value equal to . . Numerical eval-uation for each of these cases gives rise to the minimum values as shown intable 2. Thus the optimal dual value is . and this is strictly greater thanthe primal value. Theorem 4.4.
The primal problem (I) is NP-hard.
212 D2C1 . . D1 . . Figure 2: The numbers in the cells denote the optimum value of the objectivefunction (VII) in the corresponding cases.
Proof.
We describe a polynomial time procedure that reduces an instance ofthe integer partition problem to a special case of the primal problem. Givena set of positive integers { c , c , . . . , c n } , the integer partition problem is tofind a subset S ⊂ [ n ] , such that X i ∈ S c i = X i/ ∈ S c i . If such a set S exists, then we say that an integer partition exists. Consider anetwork with n players and n + 1 link constraints given by y i ≤ c i , ∀ i ∈ [ n ] , and n X i =1 y i ≤ P ni =1 c i . It is easy to realize a network with these link constraints. Let k = 2 . Let theCPT characteristics of all the players be as follows: h i (1) = 1 − ǫ, h i (2) = ǫ, v i ( x i ) = x i , ∀ i ∈ [ n ] , where ǫ = 1 / . Let W ps denote the optimal value of the system problem. Weshow that W ps ≥ T := (1 − ǫ ) P i ∈ [ n ] c i if and only if an integer partition exists.Suppose an integer partition exists and is given by the set S , consider theallocation π i = [1 , if i ∈ S and π i = [2 , otherwise, z i (1) = c i , z i (2) = 0 for all i ∈ [ n ] . The aggregate utility for this allocation is equal to T and hence W ps ≥ T . Suppose W ps ≥ T . Then there an allocation, say z ∗ and π ∗ withaggregate utility at least T . Since k = 2 , π ∗ actually defines a partition of [ n ] , given by S = { i ∈ [ n ] : π i (1) = 1 } . We have, the aggregate utility W (1) + W (2) ≥ T, where W (1) := X i ∈ S (1 − ǫ ) z i (1) + X i/ ∈ S ǫz i (2) ,W (2) := X i/ ∈ S (1 − ǫ ) z i (1) + X i ∈ S ǫz i (2) . W (1) and W (2 is at least as big as T / . Without loss ofgenerality, let W (1) ≥ T / . Thus we have, X i ∈ S z i (1) + ǫ − ǫ X i/ ∈ S z i (2) ≥ P i ∈ [ n ] c i . However, since z ∗ is feasible, the link constraints give X i ∈ S z i (1) + X i/ ∈ S z i (2) ≤ P i ∈ [ n ] c i . Since ǫ < / , we should have P i/ ∈ S z i (2) = 0 and P i ∈ S z i (1) = ( P i ∈ [ n ] c i ) / ,implying that S forms an integer partition. This completes the proof. Suppose it is enough to ensure that the link constraints are satisfied in ex-pectation, as in the average system problem. Consider the function V avg i (¯ z i ) on R + given by the value of the following optimization problem:Maximize k X l =1 h i ( l ) v i ( z i ( l )) subject to k k X l =1 z i ( l ) = ¯ z i ,z i ( l ) ≥ z i ( l + 1) , ∀ l ∈ [ k ] . (VIII)Let Z i (¯ z i ) denote the set of feasible ( z i ( l )) l ∈ [ k ] in the above problem forany fixed ¯ z i ≥ . We observe that Z i (¯ z i ) is a closed and bounded polytope,and hence V avg i (¯ z i ) is well defined. Lemma 5.1.
For any continuous, differentiable, concave and strictly increasingvalue function v i ( · ) , the function V avg i ( · ) is continuous, differentiable, concaveand strictly increasing in ¯ z i . We prove this lemma in Appendic A. The average system problem SYS_AVG [ z ; h, v, A, c ] n X i =1 V avg i (¯ z i ) subject to X i ∈ R j ¯ z i ≤ c j , ∀ j, ¯ z i ≥ , ∀ i. Kelly [13] showed that this problem can be decomposed into user problems,one for each user i , Maximize V avg i (¯ z i ) − ¯ ρ i ¯ z i subject to ¯ z i ≥ , and a network problem, Maximize n X i =1 ¯ ρ i ¯ z i subject to X i ∈ R j ¯ z i ≤ c j , ∀ j, ¯ z i ≥ , ∀ i, in the sense that there exist ¯ ρ i ≥ , ∀ i ∈ [ n ] , such that the optimum solutions ¯ z i of the user problems, for each i , solve the network problem and the aver-age system problem. Note that this decomposition is different from the onepresented in Section 3. Here the network problem aims at maximizing itstotal revenue P ni =1 ¯ ρ i ¯ z i , instead of maximizing a weighted aggregate utilitywhere the utility is replaced with a proxy logarithmic function. The abovedecomposition is not as useful as the decomposition in Section 3 in order todevelop iterative schemes that converge to equilibrium. However, the abovedecomposition motivates the following user problem:USER_AVG [ z i ; ¯ ρ i , h i , v i ] Maximize k X l =1 h i ( l ) v i ( z i ( l )) − ¯ ρ i k k X l =1 z i ( l ) subject to z i ( l ) ≥ z i ( l + 1) , ∀ l ∈ [ k ] , where, as before, z i ( k + 1) = 0 .We observed in Proposition 4.2 that strong duality holds in the averagesystem problem. Let z ∗ be the optimum lottery scheme that solves this prob-lem. Then, first of all, z ∗ satisfies z ∗ i ( l ) ≥ z ∗ i ( l + 1) ∀ i, l and is feasible in24xpectation, i.e., ¯ z ∗ := (¯ z ∗ i ) i ∈ [ n ] ∈ F , where ¯ z ∗ i := (1 /k ) P l z ∗ i ( l ) . Further, z ∗ optimizes the objective function of the average system problem. Besides,there exist ¯ λ ∗ j ≥ for all j such that the primal average problem (V) and thedual average problem (VI) each attain their optimum at z ∗ , (¯ λ ∗ j , j ∈ [ m ]) .For player i , consider the price ¯ ρ ∗ i := P j ∈ J i ¯ λ ∗ j , which is obtained by sum-ming the prices ¯ λ ∗ j corresponding to the links on player i ’s route. From thedual average problem (VI), fixing ¯ λ j = ¯ λ ∗ j ∀ j , we get that the optimum lotteryallocation z ∗ i for player i should optimize the problem USER_AVG [ z i ; ¯ ρ ∗ i , h i , v i ] .We now impose some additional conditions on the probability weightingfunction that are typically assumed based on empirical evidence and certainpsychological arguments [12]. We assume that the probability weightingfunction w i ( p i ) is concave for small values of the probability p i and convexfor the rest. Formally, there exists a probability ˜ p i ∈ [0 , such that w i ( p i ) is concave over the interval p i ∈ [0 , ˜ p i ] and convex over the interval [˜ p i , .Typically the point of inflection, ˜ p i , is around / .Let w ∗ i : [0 , → [0 , be the minimum concave function that dominates w i ( · ) , i.e., w ∗ i ( p i ) ≥ w i ( p i ) for all p i ∈ [0 , . Let p ∗ i ∈ [0 , be the smallestprobability such that w ∗ i ( p i ) is linear over the interval [ p ∗ i , . Lemma 5.2.
Given the assumptions on w i ( · ) , we have p ∗ i ≤ ˜ p i and w ∗ i ( p i ) = w i ( p i ) for p i ∈ [0 , p ∗ i ] . If p ∗ i < , then for any p i ∈ [ p ∗ i , , we have w i ( p i ) ≤ w i ( p i ) + ( p i − p i ) 1 − w i ( p i )1 − p i . (5.1) for all p i ∈ [ p i , . A proof of this lemma is included in Appendix A. We now show that,under certain conditions, the optimal lottery allocation z ∗ i satisfies z ∗ i ( l ∗ ) = z ∗ i ( l ∗ + 1) = · · · = z ∗ i ( k ) , (5.2)where l ∗ := min { l ∈ [ k ] : ( l − /k ≥ p ∗ i } , provided p ∗ i ≤ ( k − /k . As aresult, for a typical optimum lottery allocation, the lowest allocation occurswith a large probability approximately equal to − p ∗ i , and with a few higherallocations that we recognize as bonuses. Proposition 5.3.
For any average user problem USER_AVG [ z i ; ¯ ρ ∗ i , h i , v i ] with astrictly increasing, continuous, differentiable and strictly concave value function v i ( · ) , and a strictly increasing continuous probability weighting function w i ( · ) (satisfying w i (0) = 0 and w i (1) = 1 ) such that p ∗ i ≤ ( k − /k , the optimumlottery allocation z ∗ i satisfies Equation (5.2) . roof. The Lagrangian for the average user problem USER_AVG [ z i ; ¯ ρ ∗ i , h i , v i ] is L ( z i ; α i ) = k X l =1 h i ( l ) v i ( z i ( l )) − ¯ ρ ∗ i k k X l =1 z i ( l ) + k X l =1 α i ( l )[ z i ( l ) − z i ( l + 1)] , where α i ( l ) ≥ are the dual variables corresponding to the order constraints z i ( l ) ≥ z i ( l + 1) , and α i (0) = 0 . Differentiating with respect to z i ( l ) , we get, ∂ L ( z i ; α i ) ∂z i ( l ) = h i ( l ) v ′ i ( z i ( l )) − ¯ ρ ∗ i k + α i ( l ) − α i ( l − . Since the problem USER_AVG [ z i ; ¯ ρ ∗ i , h i , v i ] has a concave objective functionand linear constraints, there exist α ∗ i ( l ) ≥ such that h i ( l ) v ′ i ( z ∗ i ( l )) = ¯ ρ ∗ i k − α ∗ i ( l ) + α ∗ i ( l − , ∀ l ∈ [ k ] , (5.3)and α ∗ i ( l )[ z ∗ i ( l ) − z ∗ i ( l + 1)] = 0 , ∀ l ∈ [ k ] . (5.4)If z ∗ i consists of identical allocations then it trivially satisfies Equation (5.2).If not, then there exists l ∈ { , . . . , k } such that z ∗ i ( l − > z ∗ i ( l ) = z ∗ i ( l + 1) = · · · = z ∗ i ( k ) , i.e. z ∗ i ( l ) is the lowest allocation and occurs with probability ( k − l + 1) /k ,and the next lowest allocation is equal to z ∗ i ( l − . Summing the equationscorresponding to l ≤ l ≤ k from (5.3), we get " k X s = l h i ( s ) v ′ i ( z ∗ i ( l )) = (cid:18) k − l + 1 k (cid:19) ¯ ρ ∗ i − α ∗ i ( k ) + α ∗ i ( l − . (5.5)The equation corresponding to l = l − in (5.3) says, h i ( l − v ′ i ( z ∗ i ( l − ρ ∗ i k − α ∗ i ( l −
1) + α ∗ i ( l − . (5.6)Since z ∗ i ( l − > z ∗ i ( l ) , from (5.4), we have α ∗ i ( l −
1) = 0 . Thus from(5.5) and (5.6), we have h i ( l − v ′ i ( z ∗ i ( l − ≥ ¯ ρ ∗ i k ≥ k − l + 1 " k X s = l h i ( s ) v ′ i ( z ∗ i ( l )) . v i ( · ) is strictly concave and strictly increasing, z ∗ i ( l − >z ∗ i ( l ) implies < v ′ i ( z ∗ i ( l − < v ′ i ( z ∗ i ( l )) . Thus, h i ( l − > k − l + 1 " k X s = l h i ( s ) . (5.7)If ( l − /k ≥ p ∗ i , then h i ( l −
1) = w i (cid:18) l − k (cid:19) − w i (cid:18) l − k (cid:19) ≤ k − l + 1 (cid:20) w i (1) − w i (cid:18) l − k (cid:19)(cid:21) = 1 k − l + 1 " k X s = l h i ( s ) . (5.8)where the inequality follows from (5.1) with p i = ( l − /k and p i = ( l − /k . However, (5.8) contradicts (5.7) and hence ( l − /k < p ∗ i . This provesthe lemma. We saw that if we take the probabilistic sensitivity of players into account,then lottery allocation improves the ex ante aggregate utility of the players.We considered the RDU model, a special case of CPT utility, to model prob-abilistic sensitivity. This model, however, is restricted to reward allocations,and it would be interesting to extend it to a general CPT model with refer-ence point and loss aversion. This will allow us to study loss allocations asin punishment or burden allocations, for example criminal justice, militarydrafting, etc.For any fixed permutation profile, we showed the existence of equilibriumprices in a market-based mechanism to implement an optimal lottery. Wealso saw that finding the optimal permutation profile is an NP-hard problem.We note that the system problem has parallels in cross-layer optimization inwireless [16] and multi-route networks [30]. Several heuristic methods havehelped achieve approximately optimal solutions in cross-layer optimization.Similar methods need to be developed for our system problem. We leave thisfor future work.The hardness in the system problem comes from hard link constraints.Hence, by relaxing these conditions to hold only in expectation, we derived27ome qualitative features of the optimal lottery structure under the typicalassumptions on the probability weighting function of each agent in the RDUmodel. As observed, the players typically ensure their minimum allocationwith high probability, and gamble for higher rewards with low probability.
A Proofs
Proof of Lemma 4.1.
Suppose problem (I) and its dual (II) have the samevalue. The value of (I) is same as that of the system problem SYS [ z, π ; h, v, A, c ] .Let us denote the objective function by Θ( π, z, λ ) := n X i =1 k X l =1 h i ( l ) v i ( z i ( l ))+ m X j =1 k X l =1 λ j ( l ) c j − X i ∈ R j z i ( π i ( l )) . Since F is a polytope, for any fixed permutation profile π , the set of feasible z is closed and bounded. The function Θ( π, z, λ ) is continuous in z and hencethe fixed-permutation system problem SYS_FIX [ z ; π, h, v, A, c ] has a boundedvalue. We also note that this value is non-negative. Since there are finitelymany permutation profiles π ∈ Q i S k , maximizing over these, we get thatthe system problem has a bounded non-negative value, say W , achieved sayat z ∗ and π ∗ . Thus n X i =1 k X l =1 h i ( l ) v i ( z ∗ i ( l )) = W, (A.1)and the lottery z ∗ is feasible with respect to the permutation profile π ∗ , i.e. z ∗ i ( l ) ≥ z ∗ i ( l + 1) for all i ∈ [ n ] , l ∈ [ k ] and X i ∈ R j z ∗ i ( π ∗ i ( l )) ≤ c j for all j ∈ [ m ] , l ∈ [ k ] . (A.2)If this were not true, then the minimum of the objective function Θ( π ∗ , z ∗ , λ ) with respect to λ would be −∞ and not W ≥ .The value of the dual problem (II) is equal to W . Consider the function Θ d : R m × k + → R , given by maximizing over the objective function in problem(II), with respect to π and z for a fixed λ ≥ , Θ d ( λ ) := max π i ∈ S k ∀ i,z : z i ( l ) ≥ z i ( l +1) ∀ i,l Θ( π, z, λ ) .
28e note that the function Θ d ( λ ) is lower semi-continuous, since the function Θ( π, z, λ ) is continuous in λ . Since ( v i ( · ) , ∀ i ) are concave strictly increas-ing functions, there exists a sufficiently large finite λ such that ≤ M :=Θ d ( λ ) < ∞ . It follows that the minimum of Θ d ( λ ) is achieved over the do-main defined by λ j ( l ) ∈ [0 , M/ (min j c j )] for all j, l . Since this is a boundedregion and the function Θ d ( λ ) is lower semi-continuous, there exists a λ ∗ such that Θ d ( λ ∗ ) = min λ ≥ Θ d ( λ ) = W .Since Θ d ( λ ∗ ) = W , we have Θ( π ∗ , z ∗ , λ ∗ ) ≤ W . However, from (A.1),(A.2) and the fact that λ ∗ j ( l ) ≥ for all j, l we get Θ( π ∗ , z ∗ , λ ∗ ) ≥ W . Hence Θ( π ∗ , z ∗ , λ ∗ ) = W . Thus the maximum in the definition of Θ d ( λ ∗ ) is achievedat z ∗ , π ∗ . This implies π ∗ i satisfies (4.1) for all i . Proof of Lemma 5.1.
Let ¯ z i ≥ and τ > . Let z ∗ i ∈ Z i (¯ z i ) be such that V avg i (¯ z i ) = P kl =1 h i ( l ) v i ( z ∗ i ( l )) . We have, ( z ∗ i ( l ) + τ ) l ∈ [ k ] ∈ Z i (¯ z i + τ ) and V avg i (¯ z i + τ ) ≥ k X l =1 h i ( l ) v i ( z ∗ i ( l ) + τ ) > k X l =1 h i ( l ) v i ( z ∗ i ( l )) = V avg i (¯ z i ) , where the strict inequality follows from the fact that v i ( · ) is strictly increas-ing. This establishes that the function V avg i (¯ z i ) is strictly increasing.Let ¯ z i , ¯ z i ≥ and σ ∈ [0 , . Let z i ∈ Z i (¯ z i ) and z i ∈ Z i (¯ z i ) be suchthat V avg i (¯ z i ) = P kl =1 h i ( l ) v i ( z i ( l )) and V avg i (¯ z i ) = P kl =1 h i ( l ) v i ( z i ( l )) . Let z σi := σz i ( l ) + (1 − σ ) z i ( l ) and ¯ z σi := σ ¯ z i ( l ) + (1 − σ )¯ z i ( l ) . Then z σi ∈ Z i (¯ z σi ) and by the concavity of v i ( · ) , we have V avg i (¯ z σi ) ≥ k X l =1 h i ( l ) v i ( z σi ( l )) ≥ k X l =1 h i ( l ) (cid:2) σv i ( z i ( l )) + (1 − σ ) v i ( z i ( l )) (cid:3) = σV avg i (¯ z i ) + (1 − σ ) V avg i (¯ z i ) . This establishes that the function V avg i (¯ z i ) is concave. This implies that V avg i (¯ z i ) is continuous and directionally differentiable at each ¯ z i > , and we have thefollowing relation between its left and right directional derivatives (see, forexample, [24]): dd ¯ z i V avg i (¯ z i − ) ≥ dd ¯ z i V avg i (¯ z i +) , for all ¯ z i > . (A.3)Further, if (¯ z ti ) t ≥ is a sequence such that ¯ z ti → , and z ti ∈ Z i (¯ z ti ) for all t ≥ ,then z ti ( l ) → for all l ∈ [ k ] . By the continuity of the function v i ( · ) , we havethat the function V avg i (¯ z i ) is continuous at ¯ z i = 0 .29et ¯ z i > , and let τ t := 1 /t , for t ≥ . As before, let z ∗ i ∈ Z i (¯ z i ) be suchthat V avg i (¯ z i ) = P kl =1 h i ( l ) v i ( z ∗ i ( l )) . Since ¯ z i > and (1 /k ) P kl =1 z ∗ i ( l ) = ¯ z i ,we have z ∗ i ( l ) > z ∗ i ( l + 1) for at least one l ∈ [ k ] . Let ˆ l ∈ [ k ] be the smallestsuch l . For t ≥ , let z t + i be given by z t + i ( l ) := ( z ∗ i ( l ) + k ˆ l τ t , for ≤ l ≤ ˆ l,z ∗ i ( l ) , for ˆ l < l ≤ k. Note that z t + i ∈ Z i (¯ z i + τ t ) . We have, V avg i (¯ z i + τ t ) − V avg i (¯ z i ) − kτ t ˆ l ˆ l X l =1 h i ( l ) v ′ i ( z ∗ i ( l )) ≥ k X l =1 h i ( l ) v i ( z t + i ( l )) − k X l =1 h i ( l ) v i ( z ∗ i ( l )) − kτ t ˆ l ˆ l X l =1 h i ( l ) v ′ i ( z ∗ i ( l ))= ˆ l X l =1 h i ( l ) (cid:20) v i ( z ∗ i ( l ) + kτ t / ˆ l ) − v i ( z ∗ i ( l )) − kτ t ˆ l v ′ i ( z ∗ i ( l )) (cid:21) . Let γ ∗ := k ˆ l P ˆ ll =1 h i ( l ) v ′ i ( z ∗ i ( l )) . We have, dd ¯ z i V avg i (¯ z i +) ≥ lim inf t →∞ V avg i (¯ z i + τ t ) − V avg i (¯ z i ) τ t ≥ γ ∗ . (A.4)Similarly, for t ≥ ⌈ k/z ∗ i (ˆ l ) ⌉ (here, ⌈·⌉ denotes the ceiling function), let z t − i be given by z t − i ( l ) := ( z ∗ i ( l ) − k ˆ l τ t , for ≤ l ≤ ˆ l,z ∗ i ( l ) , for ˆ l < l ≤ k. We observe that z t − i ∈ Z i (¯ z i − τ t ) , for all t ≥ ⌈ k/z ∗ i (ˆ l ) ⌉ , and we have V avg i (¯ z i ) − V avg i (¯ z i − τ t ) − kτ t ˆ l ˆ l X l =1 h i ( l ) v ′ i ( z ∗ i ( l )) ≤ k X l =1 h i ( l ) v i ( z ∗ i ( l )) − k X l =1 h i ( l ) v i ( z t − i ( l )) − kτ t ˆ l ˆ l X l =1 h i ( l ) v ′ i ( z ∗ i ( l ))= ˆ l X l =1 h i ( l ) (cid:20) v i ( z ∗ i ( l )) − v i ( z ∗ i ( l ) − kτ t / ˆ l ) − kτ t ˆ l v ′ i ( z ∗ i ( l )) (cid:21) . dd ¯ z i V avg i (¯ z i − ) ≤ lim sup t →∞ V avg i (¯ z i ) − V avg i (¯ z i − τ t ) τ t ≤ γ ∗ . (A.5)From (A.3), (A.4) and (A.5), we have dd ¯ z i V avg i (¯ z i − ) = dd ¯ z i V avg i (¯ z i +) = γ ∗ . (A.6)This establishes that the function V avg i (¯ z i ) is differentiable and completes theproof. Proof of Lemma 5.2.
Consider the function ¯ w i : [0 , → [0 , , given by ¯ w i ( p i ) := ( w ∗ i ( p i ) for ≤ p i < ˜ p i ,w ∗ i (˜ p i ) + ( p i − ˜ p i ) − w ∗ i (˜ p i )1 − ˜ p i for ˜ p i ≤ p i ≤ . Since w ∗ i is concave on [0 , , one can verify that the function ¯ w i is also con-cave on [0 , . Since w ∗ i dominates w i , we have w ∗ i (˜ p i ) ≥ w i (˜ p i ) . Since thefunction w i ( p i ) is convex on the interval [˜ p i , , we have w i ( p i ) ≤ ¯ w i ( p i ) , for p i ∈ [˜ p i , . However, since w ∗ i is the minimum concave function that dom-inates w i , we get ¯ w i = w ∗ i . Thus, w ∗ i is linear over the interval [˜ p i , , andhence p ∗ i ≤ ˜ p i .Suppose ˜ p i = 1 . Then w i ( · ) is a concave function on the unit interval [0 , ,and hence w ∗ i ( p i ) = w i ( p i ) , for p i ∈ [0 , ⊃ [0 , p ∗ i ] . Further, if p ∗ i < , then w i ( p i ) is linear over [ p ∗ i , , and inequality (5.1) holds, in fact, with equality.This completes the proof of Lemma 5.2, if ˜ p i = 1 .For the rest of the proof we assume ˜ p i < . Define g i : [0 , → R + as g i ( p i ) := 1 − w i ( p i )1 − p i . We now provide an alternate characterization of the function w ∗ i and thepoint p ∗ i . Let ˆ p i ∈ [0 , be given by ˆ p i := min arg min p i ∈ [0 , ˜ p i ] { g ( p i ) } . The existence of ˆ p i is guaranteed by the continuity of the function g i ( p i ) onthe compact interval [0 , ˜ p i ] . Let ˆ a i := g i (ˆ p i ) . Since w i ( p i ) is convex overthe interval [˜ p i , , the function g i ( p i ) is nondecreasing over [˜ p i , . Hence31 i ( p i ) ≥ g i (ˆ p i ) , for p i ∈ [0 , . Substituting the expression for g i ( · ) and rear-ranging, we get w i ( p i ) ≤ w i (ˆ p i ) + ˆ a i ( p i − ˆ p i ) , for p i ∈ [0 , . Since the function w i ( p i ) is concave on the interval [0 , ˆ p i ] andthe linear function w i (ˆ p i ) + ˆ a i ( p i − ˆ p i ) dominates w i ( p i ) on [0 , , we havethat the following function ˆ w i ( p i ) is concave on [0 , : ˆ w i ( p i ) := ( w i ( p i ) for ≤ p i < ˆ p i ,w i (ˆ p i ) + ˆ a i ( p i − ˆ p i ) for ˆ p i ≤ p i ≤ . It follows that w ∗ i ( p i ) = ˆ w i ( p i ) for p i ∈ [0 , . Thus, p ∗ i ≤ ˆ p i and w ∗ i ( p i ) =ˆ w i ( p i ) = w i ( p i ) for p i ∈ [0 , p ∗ i ] . If ˆ p i = 0 , then p ∗ i = ˆ p i . If ˆ p i > , then fromthe definition of ˆ p i , we have g i ( p i ) > g i (ˆ p i ) for p i ∈ [0 , ˆ p i ) , and this impliesthat ˆ p i = p ∗ i .We now prove inequality (5.1). Since ˜ p i < we have p ∗ i < . Rearrang-ing we get that inequality (5.1) is equivalent to showing that the function g i ( p i ) is non-decreasing over the interval [ p ∗ i , . As observed earlier, g i ( p i ) isnon-decreasing over the interval [˜ p i , . Hence it is enough to show that thefunction g i ( p i ) is non-decreasing on [ p ∗ i , ˜ p i ] . Suppose, on the contrary, thereexist p i , p i ∈ [ p ∗ i , ˜ p i ] such that p i < p i and g i ( p i ) > g i ( p i ) . Since p ∗ i = ˆ p i , andfrom the definition of ˆ p i , we have p i > p ∗ i and g i ( p ∗ i ) ≤ g i ( p i ) . Since, g i ( p i ) is a continuous function, there exist p i ∈ [ p ∗ i , p i ) such that g i ( p i ) = g i ( p i ) .Thus we have p i < p i < p i such that g i ( p i ) > g i ( p i ) = g i ( p i ) . However, thiscontradicts the concavity of w i ( p i ) on [ p ∗ i , ˜ p i ] . This completes the proof. References [1] E. Altman and L. Wynter. Equilibrium, games, and pricing in trans-portation and telecommunication networks.
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